Ogg's Torsion conjecture: Fifty years later

TL;DR

This paper surveys the fifty-year history of Ogg's Torsion Conjecture, exploring the existence of rational points on modular curves and their geometric significance in elliptic curve theory.

Contribution

It provides a comprehensive overview of the developments related to Ogg's Torsion Conjecture and its impact on understanding rational points on modular curves.

Findings

Progress in understanding rational points on modular curves

Connections between geometric reasons and existence of rational points

Development of new results and conjectures inspired by Ogg's work

Abstract

Andrew Ogg's mathematical viewpoint has inspired an increasingly broad array of results and conjectures. His results and conjectures have earmarked fruitful turning points in our subject, and his influence has been such a gift to all of us. Ogg's celebrated Torsion Conjecture -- as it relates to modular curves -- can be paraphrased as saying that rational points (on the modular curves that parametrize torsion points on elliptic curves) exist if and only if there is a good geometric reason for them to exist. We give a survey of Ogg's Torsion Conjecture and the subsequent developments in our understanding of rational points on modular curves over the last fifty years.